Kazhdan--Lusztig Polynomials for permutations with $P_{\mathrm{id},w}(1)=3$ and Irreducible Singular Locus

We determine the Kazhdan--Lusztig polynomial $P_{\mathrm{id},w}(q)$ for permutations $w$ such that $P_{\mathrm{id},w}(1)=3$ and the Schubert variety $X_w$ has irreducible singular locus. Let $h=h(w)$ be the minimum height $w(a)-w(d)$ of a $3412$-occurrence in $w$ at positions $a<b<c<d$, with $h=1$ w